REGULAR LOW-DENSITY PARITY-CHECK CODES BASED ON SHIFTED IDENTITY MATRICES

This paper extends the class of Low-Density Parity-Check (LDPC) codes that can be constructed from shifted identity matrices. To construct regular LDPC codes, a new method is proposed. Two simple inequations are adopted to avoid the short cycles in Tanner graph, which makes the girth of Tanner graphs at least 8. Because their parity-check matrices are made up of circulant matrices, the new codes are quasi-cyclic codes. They perform well with iterative decoding.

THE Q-MATRIX LOW-DENSITY PARITY-CHECK CODES

This paper presents a matrix permuting approach to the construction of Low-Density Parity-Check (LDPC) code. It investigates the structure of the sparse parity-check matrix defined by Gallager. It is discovered that the problem of constructing the sparse parity-check matrix requires an algorithm that is efficient in search environments and also is able to work with constraint satisfaction problem. The definition of Q-matrix is given, and it is found that the queen algorithm enables to search the Q-matrix. With properly permuting Q-matrix as sub-matrix, the sparse parity-check matrix which satisfied constraint condition is created, and the good regular-LDPC code that is called the Q-matrix LDPC code is generated. The result of this paper is significant not only for designing low complexity encoder, improving performance and reducing complexity of iterative decoding arithmetic, but also for building practical system of encodable and decodable LDPC code.

CONSTRUCTION OF NONSYSTEMATIC LOW-DENSITY PARITY-CHECK CODES BASED ON SYMMETRIC BALANCED INCOMPLETE BLOCK DESIGN

This paper studies the nonsystematic Low-Density Parity-Check(LDPC)codes based onSymmetric Balanced Incomplete Block Design(SBIBD).First,it is concluded that the performancedegradation of nonsystematic linear block codes is bounded by the average row weight of generalizedinverses of their generator matrices and code rate.Then a class of nonsystematic LDPC codes con-structed based on SBIBD is presented.Their characteristics include:both generator matrices andparity-check matrices are sparse and cyclic,which are simple to encode and decode;and almost arbi-trary rate codes can be easily constructed,so they are rate-compatible codes.Because there aresparse generalized inverses of generator matrices,the performance of the proposed codes is only0.15dB away from that of the traditional systematic LDPC codes.

Low-Complexity Optimization Algorithm for Irregular Low-Density Parity-Check Codes

A low-complexity algorithm is proposed in this paper in order to optimize irregular low-density parity-check(LDPC)codes.The algorithm proposed can calculate the noise threshold by means of a one-dimensional density evolution and search the optimal degree profiles with fast-convergence differential evolution,so that it has a lower complexity and a faster convergence speed.Simulation results show that the irregular LDPC codes optimized by the presented algorithm can also perform better than Turbo codes at moderate block length even with less computation cost.

Low-Density Parity-Check Codes: Research Status and Development Direction

In this paper, we conclude five kinds of methods for construction of the regular low-density parity matrix H and three kinds of methods for the construction of irregular low-density parity-check matrix H. Through the analysis of the code rate and parameters of these eight kinds of structures, we find that the construction of low-density parity-check matrix tends to be more flexible and the parameter variability is enhanced. We propose that the current development cost should be lower with the progress of electronic technology and we need research on more practical Low-Density Parity-Check Codes (LDPC). Combined with the application of the quantum distribution key, we urgently need to explore the research direction of relevant theories and technologies of LDPC codes in other fields of quantum information in the future.

PERFORMANCE EVALUATION OF LOW DENSITY PARITY CHECK CODES FOR DIGITAL RADIO MONDIALE (DRM) SYSTEM

In Digital Radio Mondiale (DRM) system, achieving good audio quality becomes a challenge due to its limited band-width of 9 or 10kHz and the very bad fading channels. Therefore, DRM needs highly efficient channel coding schemes. This paper, proposes the schemes which use the Low-Density Parity-Check (LDPC) coded Bit-Interleaved Coded Modulation (BICM) schemes for the implementation of DRM systems. Simulation results show that the proposed system is more efficient than the Rate Compatible Punctured Con volutional (RCPC) coded DRM system on various broadcast channels, and may be recommended as a coding technology for Digital Amplitude Modulation Broadcasting (DAMB) systems of China.

DESIGN OF LOW-DENSITY PARITY CHECK CODES FOR MIMO SYSTEMS WITH DOUBLY-ITERATIVE RECEIVERS

In this paper, the Multiple Input Multiple Output (MIMO) doubly-iterative receiver which consists of the Probabilistic Data Association detector (PDA) and Low-Density Parity-Check Code (LDPC) decoder is developed. The receiver performs two iterative decoding loops. In the outer loop, the soft information is exchanged between the PDA detector and the LDPC decoder. In the inner loop, it is exchanged between variable node and check node decoders inside the LDPC decoder. On the light of the Extrinsic Information Transfer (EXIT) chart technique, an LDPC code degree profile optimization algorithm is developed for the doubly-iterative receiver. Simulation results show the doubly-receiver with optimized irregular LDPC code can have a better performance than the one with the regular one.

High Robust Broadcasting over DTMB-A with Low-rate LDPC Codes

As the 2nd generation digital terrestrial television broadcasting(DTTB)standard,digital terrestrial/television multimedia broadcasting-advanced(DTMB-A)can provide higher spectrum efficiency and transmission reliability by adopting flexible frame structure and advanced forward error correction coding compared with the 1 st generation DTTB systems.In order to increase the flexibility and robustness of the DTTB network,the frequency reuse scheme of factor one(reuse-1)is proposed,where the same RF channel is used by different stations covering the adjacent service areas.However,it demands a very low carrier-tonoise ratio(C/N)threshold below 0 dB at the DTTB physical layer.In this paper,a robust broadcasting technique is proposed based on DTMB-A with newly designed low-rate low density parity check(LDPC)codes.By adopting quasi-cyclic(QC)Raptor-like structure and progressive lifting method,the high performance low-rate LDPC codes are designed supporting multiple code lengths.Both density-evolution analyses and laboratory measurements demonstrate that DTMB-A with low-rate coding can complete the demodulation reliably with the C/N threshold below0 d B,which is one important necessary condition to support frequency reuse-1 scheme.

Construction of protograph LDPC codes with circular generator matrices

The application of protograph low density parity check （LDPC） codes involves the encoding complexity problem. Since the generator matrices are dense, and if the positions of ＂1＂ s are irregularity, the encoder needs to store every ＂1＂ of the generator matrices by using huge chip area. In order to solve this problem, we need to design the protograph LDPC codes with circular generator matrices. A theorem concerning the circulating property of generator matrices of nonsingular protograph LDPC codes is proposed. The circulating property of generator matrix of nonsingular protograph LDPC codes can be obtained from the corresponding quasi-cyclic parity check matrix. This paper gives a scheme of constructing protograph LDPC codes with circulating generator matrices, and it reveals that the fast encoding algorithm of protograph LDPC codes has lower encoding complexity under the condition of the proposed theorem. Simulation results in ad- ditive white Gaussian noise （AWGN） channels show that the bit error rate （BER） performance of the designed codes based on the proposed theorem is much better than that of GB20600 LDPC codes and Tanner LDPC codes.

Construction of time-frequency codes based on protograph LDPC codes in OFDM communication systems

This paper proposes a scheme to construct time- frequency codes based on protograph low density parity check （LDPC） codes in orthogonal frequency division multiplexing （OFDM） communication systems. This approach synthesizes two techniques： protograph LDPC codes and OFDM. One symbol of encoded information by protograph LDPC codes corresponds to one sub-carrier, namely the length of encoded information equals to the number of sub-carriers. The design of good protograph LDPC codes with short lengths is given, and the proposed proto- graph LDPC codes can be of fast encoding, which can reduce the encoding complexity and simplify encoder hardware implementa- tion. The proposed approach provides a higher coding gain in the Rayleigh fading channel. The simulation results in the Rayleigh fading channel show that the bit error rate （BER） performance of the proposed time-frequency codes is as good as random LDPC- OFDM codes and is better than Tanner LDPC-OFDM codes under the condition of different fading coefficients.

Performance analysis of LDPC codes on OOK terahertz wireless channels

Atmospheric absorption, scattering, and scintillation are the major causes to deteriorate the transmission quality of terahertz（THz） wireless communications. An error control coding scheme based on low density parity check（LDPC） codes with soft decision decoding algorithm is proposed to improve the bit-error-rate（BER） performance of an on-off keying（OOK） modulated THz signal through atmospheric channel. The THz wave propagation characteristics and channel model in atmosphere is set up. Numerical simulations validate the great performance of LDPC codes against the atmospheric fading and demonstrate the huge potential in future ultra-high speed beyond Gbps THz communications.

Design of good QC-LDPC codes without small girth in the p-plane

A construction method based on the p-plane to design high-girth quasi-cyclic low-density parity-check （QC-LDPC） codes is proposed. Firstly the good points in every line of the p-plane can be ascertained through filtering the bad points, because the designed parity-check matrixes using these points have the short cycles in Tanner graph of codes. Then one of the best points from the residual good points of every line in the p-plane will be found, respectively. The optimal point is also singled out according to the bit error rate （BER） performance of the QC-LDPC codes at last. Explicit necessary and sufficient conditions for the QC-LDPC codes to have no short cycles are presented which are in favor of removing the bad points in the p-plane. Since preventing the short cycles also prevents the small stopping sets, the proposed construction method also leads to QC-LDPC codes with a higher stopping distance.

DESIGN OF QUASI-CYCLIC LDPC CODES BASED ON EUCLIDEAN GEOMETRIES

A new method for constructing Quasi-Cyclic (QC) Low-Density Parity-Check (LDPC) codes based on Euclidean Geometry (EG) is presented. The proposed method results in a class of QC-LDPC codes with girth of at least 6 and the designed codes perform very close to the Shannon limit with iterative decoding. Simulations show that the designed QC-LDPC codes have almost the same performance with the existing EG-LDPC codes.

Weighted symbol-flipping decoding algorithm for nonbinary LDPC codes with flipping patterns

A novel low-complexity weighted symbol-flipping algorithm with flipping patterns to decode nonbinary low-density parity-check codes is proposed. The proposed decoding procedure updates the hard-decision received symbol vector iteratively in search of a valid codeword in the symbol vector space. Only one symbol is flipped in each iteration, and symbol flipping function, which is employed as the symbol flipping metric, combines the number of failed checks and the reliabilities of the received bits and calculated symbols. A scheme to avoid infinite loops and select one symbol to flip in high order Galois field search is also proposed. The design of flipping pattern＇s order and depth, which is dependent of the computational requirement and error performance, is also proposed and exemplified. Simulation results show that the algorithm achieves an appealing tradeoff between performance and computational requirement over relatively low Galois field for short to medium code length.

JOINT SOURCE-CHANNEL DECODING OF HUFFMAN CODES WITH LDPC CODES

In this paper, we present a Joint Source-Channel Decoding algorithm (JSCD) for Low-Density Parity Check (LDPC) codes by modifying the Sum-Product Algorithm (SPA) to account for the source redun-dancy, which results from the neighbouring Huffman coded bits. Simulations demonstrate that in the presence of source redundancy, the proposed algorithm gives better performance than the Separate Source and Channel Decoding algorithm (SSCD).

Irregular and regular LDPC codes with high spectrum efficiency modulation in image transmission over fading channel

If the degree distribution is chosen carefully, the irregular low-density parity-check (LDPC) codes can outperform the regular ones. An image transmission system is proposed by combining regular and irregular LDPC codes with 16QAM/64QAM modulation to improve both efficiency and reliability. Simulaton results show that LDPC codes are good coding schemes over fading channel in image communication with lower system complexity. More over, irregular codes can obtain a code gain of about 0.7 dB compared with regular ones when BER is 10 -4. So the irregular LDPC codes are more suitable for image transmission than the regular codes.

A NOVEL CONSTRUCTION OF QUANTUM LDPC CODES BASED ON CYCLIC CLASSES OF LINES IN EUCLIDEAN GEOMETRIES

The dual-containing (or self-orthogonal) formalism of Calderbank-Shor-Steane (CSS) codes provides a universal connection between a classical linear code and a Quantum Error-Correcting Code (QECC). We propose a novel class of quantum Low Density Parity Check (LDPC) codes constructed from cyclic classes of lines in Euclidean Geometry (EG). The corresponding constructed parity check matrix has quasi-cyclic structure that can be encoded flexibility, and satisfies the requirement of dual-containing quantum code. Taking the advantage of quasi-cyclic structure, we use a structured approach to construct Generalized Parity Check Matrix (GPCM). This new class of quantum codes has higher code rate, more sparse check matrix, and exactly one four-cycle in each pair of two rows. Ex-perimental results show that the proposed quantum codes, such as EG(2,q)II-QECC, EG(3,q)II-QECC, have better performance than that of other methods based on EG, over the depolarizing channel and decoded with iterative decoding based on the sum-product decoding algorithm.

Simulation of small size divertor tokamak plasma edge at low density of plasma

A low density plasma edge of small size divertor tokamak has been modeling by “B2SOLPS0.5.2 D” fluid transport code. The results of modeling are: 1) Formation of the strong “ITB” has detected more reliable with discovery that, low density plasma is necessary and important condition for it to form. 2) Reduction of plasma density play significantly role in the formation of the strong ITB as global parameter, possibly through change in the steep density gradient which stabilize “ITG” mode. 3) The radial electric field of small size divertor tokamak plasma edge is plasma density dependence and maximum radial electric field shear is found at low plasma density. 4) In the “NBI” discharge the toroidal (parallel) velocity at low plasma density is co-current and upward direction. 5) The structure of plasma pressure and radial electric field in quiescent H-mode are obtained.

Construction of Regular Rate-Compatible LDPC Convolutional Codes

In this paper, we propose a new method to derive a family of regular rate-compatible low-density parity-check（RC-LDPC） convolutional codes from RC-LDPC block codes. In the RC-LDPC convolutional family, each extended sub-matrix of each extended code is obtained by choosing specified elements from two fixed matrices HE1K and HE1K, which are derived by modifying the extended matrices HE1 and HE2 of a systematic RC-LDPC block code. The proposed method which is based on graph extension simplifies the design, and prevent the defects caused by the puncturing method. It can be used to generate both regular and irregular RC-LDPC convolutional codes. All resulted codes in the family are systematic which simplify the encoder structure and have maximum encoding memories which ensure the property. Simulation results show the family collectively offer a steady improvement in performance with code compatibility over binary-input additive white Gaussian noise channel（BI-AWGNC）.

SPARSE SEQUENCE CONSTRUCTION OF LDPC CODES

This letter proposes a novel and simple construction of regular Low-Density Parity-Check (LDPC) codes using sparse binary sequences. It utilizes the cyclic cross correlation function of sparse sequences to generate codes with girth8. The new codes perform well using the sumproduct decoding. Low encodingcomplexity can also be achieved due to the inherent quasi-cyclic structure of the codes.